The Hodge ring of varieties in positive characteristic

van Dobben de Bruyn, Remy

doi:10.2140/ant.2021.15.729

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The Hodge ring of varieties in positive characteristic

Remy van Dobben de Bruyn

Vol. 15 (2021), No. 3, 729–745

DOI: 10.2140/ant.2021.15.729

Abstract

Let $k$ be a field of positive characteristic. We prove that the only linear relations between the Hodge numbers $h^{i, j} (X) = dim H^{j} (X, Ω_{X}^{i})$ that hold for every smooth proper variety $X$ over $k$ are the ones given by Serre duality. We also show that the only linear combinations of Hodge numbers that are birational invariants of $X$ are given by the span of the $h^{i, 0} (X)$ and the $h^{0, j} (X)$ (and their duals $h^{i, n} (X)$ and $h^{n, j} (X)$ ). The corresponding statements for compact Kähler manifolds were proven by Kotschick and Schreieder.

Keywords

algebraic geometry, positive characteristic, Hodge cohomology, de Rham cohomology, Grothendieck ring of varieties, birational invariants

Mathematical Subject Classification 2010

Primary: 14G17

Secondary: 14A10, 14E99, 14F40, 14F99

Milestones

Received: 28 January 2020

Revised: 28 June 2020

Accepted: 7 September 2020

Published: 20 May 2021

Authors

Remy van Dobben de Bruyn
Princeton University Princeton, NJ United States	Institute for Advanced Study Princeton, NJ United States

Vol. 15, No. 3, 2021